A singular cardinal $\kappa$ is prevalent iff
$\operatorname{cov} (\kappa , \mu , \operatorname{cf} (\kappa)) = \kappa$
for some cardinal $\mu$ with
$\operatorname{cf} (\kappa) \leq \mu < \kappa$.

*Proof:*
First, it is easy to show that
$\operatorname{cov} (\kappa , \mu , \theta) \geq \kappa$
when $\theta \leq \mu < \kappa$.

Now, if $\kappa$ is prevalent, then
$\operatorname{cov} (\kappa , \mu , \operatorname{cf} (\kappa)) \leq \kappa$,
with $\eta = \operatorname{sup} \{ |A| : A \in \mathbb{A} \}$ and
$$
\mu =
\left\lbrace
\begin{array}{ll}
\eta , & \textrm{if } |A| < \eta \textrm{ for all } A \in \mathbb{A} ; \\
\eta^{+} , & \textrm{otherwise.}
\end{array}
\right.
$$
The other direction is immediate. $\square$

SSH (Shelah's Strong Hypothesis) implies SCH, hence implies PSCH.
By Corollary 3.6 in

Pierre Matet. *Large cardinals and covering numbers*,
Fundamenta Mathematicae **205** (2009), 45-75.
doi:10.4064/fm205-1-3

SSH is equivalent to the following: given cardinals $\mu$ and $\lambda$,
with $\mu \geq \lambda = \operatorname{cf} (\lambda) \geq \aleph_{1}$,
$$
\operatorname{cov} \left( \mu , \lambda , \lambda \right) =
\left\lbrace
\begin{array}{ll}
\mu , & \textrm{if } \operatorname{cf} (\mu) \geq \lambda ; \\
\mu^{+} , & \textrm{otherwise.}
\end{array}
\right.
$$

Under SSH, when $\kappa$ is a singular cardinal with
$\operatorname{cf} (\kappa) > \aleph_0$, we have, for any $\mu$ with
$\operatorname{cf} (\kappa) \leq \mu < \kappa$:
$$
\operatorname{cov} (\kappa , \mu , \operatorname{cf} (\kappa)) \leq
\operatorname{cov} (\kappa , \operatorname{cf} (\kappa) , \operatorname{cf} (\kappa)) = \kappa .
$$

Note that the condition (weaker than prevalent)
$\operatorname{cov} (\kappa , \kappa , \operatorname{cf} (\kappa)) \leq \kappa$
(for a singular cardinal $\kappa$)
is a theorem in ZFC. In fact,
$\operatorname{cov} (\kappa , \kappa , \operatorname{cf} (\kappa)) = \operatorname{cf} (\kappa)$,
by Observation 5.2(3) in Chapter II of

S. Shelah. *Cardinal Arithmetic*,
volume 29 of Oxford Logic Guides. Oxford University Press, New York, 1994.

See Definition 5.1 in Chapter II for
$\operatorname{cov} (\lambda , \kappa , \theta , \sigma)$.
Then,
$\operatorname{cov} (\lambda , \kappa , \theta) :=
\operatorname{cov} (\lambda , \kappa , \theta , 2)$.

Finally, the condition "$|B| < \operatorname{cf} (\kappa)$"
in the definition of "prevalent singular cardinal"
cannot be replaced by "$|B| \leq \operatorname{cf} (\kappa)$",
because, for any singular cardinal $\kappa$ and any $\mu$ with
$\operatorname{cf} (\kappa) < \mu < \kappa$ we have
$$
\operatorname{cov} (\kappa , \mu , {(\operatorname{cf} (\kappa))}^+) \geq
\operatorname{cov} (\kappa , \kappa , {(\operatorname{cf} (\kappa))}^+) \geq
\operatorname{cov} (\kappa , \kappa , {(\operatorname{cf} (\kappa))}^+ , \operatorname{cf} (\kappa)) \geq {\kappa}^+ ,
$$
by Fact 1 in

Andreas Liu, *Bounds for covering numbers*,
The Journal of Symbolic Logic **71** (2006), 1303-1310.
doi:10.2178/jsl/1164060456